The
classic 1935 paper of Erdos and Szekeres entitled ``A combinatorial
problem in geometry" was a starting point of a very rich discipline
within combinatorics: Ramsey theory. In that paper, Erdos and Szekeres
studied the following geometric problem. For every integer n \geq 3,
determine the smallest integer ES(n) such that any set of ES(n) points
in the plane in general position contains n members in convex position,
that is, n points that form the vertex set of a convex polygon. Their main result showed
that ES(n) \leq {2n - 4\choose n-2} + 1 = 4^{n -o(n)}. In 1960, they
showed that ES(n) \geq 2^{n-2} + 1 and conjectured this to be optimal.
Despite the efforts of many researchers, no improvement in the order of
magnitude has been made on the upper bound over the last 81 years. In
this talk, we will sketch a proof showing that ES(n) =2^{n +o(n)}.

The evolution, through spatially periodic linear dispersion, of rough
initial data leads to surprising quantized structures at rational times,
and fractal, non-differentiable profiles at irrational times. The
Talbot effect, named after an optical experiment by one of the founders
of photography, was first observed in optics and quantum mechanics, and
leads to intriguing connections with exponential sums arising in number
theory. Ramifications of these phenomena and recent progress on the
analysis, numerics, and extensions to nonlinear wave models will be
discussed.

A well-known theorem of Kuratowski (1930) in graph theory states
that a graph is planar if, and only if, it does not contain a
subdivision
of $K_5$ or $K_{3,3}$. Wagner (1937) gave a structural characterization of graphs containing no subdivision of $K_{3,3}$. Seymour in 1977 and, independently, Kelmans in 1979 conjectured that if a
graph does not contain
a subdivision of $K_5$ then it must be planar or contain a set of at most 4 vertices whose removal results in a disconnected
graph. In this talk, I will discuss additional background on this
conjecture (including connection to the Four Color Theorem), and outline
our recent proof of this conjecture (joint work with Dawei He and Yan
Wang).
I will also mention several problems that are related to this conjecture or related to our approach.

In this talk I will discuss some new applications of the
polynomial method to some classical problems in combinatorics, in
particular the Cap-Set Problem. The Cap-Set Problem is to determine the
size of the largest subset A of F_p^n having no three-term arithmetic
progressions, which are triples of vectors x,y,z satisfying x+y=2z. I will
discuss an analogue of this problem for Z_4^n and the recent progress on
it due to myself, Seva Lev and Peter Pach; and will discuss the work of
Ellenberg and Gijswijt, and of Tao, on the F_p^n version (the original
context of the problem).

Modern imaging data are often composed of several geometrically
distinct constituents. For instance, neurobiological images could
consist of a superposition of spines (pointlike objects) and
dendrites (curvelike objects) of a neuron. A neurobiologist might
then seek to extract both components to analyze their structure
separately for the study of Alzheimer specific characteristics.
However, this task seems impossible, since there are two unknowns
for every datum.
Compressed sensing is a novel research area, which was introduced in
2006, and since then has already become a key concept in various
areas of applied mathematics, computer science, and electrical
engineering. It surprisingly predicts that high-dimensional signals,
which allow a sparse representation by a suitable basis or, more
generally, a frame, can be recovered from what was previously
considered highly incomplete linear measurements, by using efficient
algorithms.
Utilizing the methodology of Compressed Sensing, the geometric
separation problem can indeed be solved both numerically and
theoretically. For the separation of point- and curvelike objects,
we choose a deliberately overcomplete representation system made of
wavelets (suited to pointlike structures) and shearlets (suited to
curvelike structures). The decomposition principle is to minimize
the $\ell_1$ norm of the representation coefficients. Our theoretical
results, which are based on microlocal analysis considerations, show
that at all sufficiently fine scales, nearly-perfect separation is
indeed achieved.
This project was done in collaboration with David Donoho (Stanford
University) and Wang-Q Lim (TU Berlin).

Effective bounds play a very important role in algebraic geometry with many applications. In this talk I will survey recent progress and open questions in the quantitative study ofreal varieties and semi-algebraic sets and their connections with other areas of mathematics -- in particular,connections to incidence geometry via the polynomial partitioning method. I will also discuss some results on the topological complexity of symmetric varieties which have a representation-theoretic flavor. Finally, if time permits I will sketch how some of these results extend to the category of constructible sheaves.

This Colloquium will be Part II of the Stelson Lecture. A function of many variables, when chosen at random, is typically
very complex. It has an exponentially large number of local minima or
maxima, or critical points. It defines a very complex landscape, the
topology of its level lines (for instance their Euler characteristic) is
surprisingly complex. This complex picture is valid even in very simple
cases, for random homogeneous polynomials of degree p larger than 2.
This has important consequences. For instance trying to find the minimum
value of such a function may thus be very difficult. The
mathematical tool suited to understand this complexity is the spectral
theory of large random matrices. The classification of the different
types of complexity has been understood for a few decades in the
statistical physics of disordered media, and in particular spin-glasses,
where the random functions may define the energy landscapes. It is also
relevant in many other fields, including computer science and Machine
learning. I will review recent work with collaborators in mathematics
(A. Auffinger, J. Cerny) , statistical physics (C. Cammarota, G. Biroli,
Y. Fyodorov, B. Khoruzenko), and computer science (Y. LeCun and his
team at Facebook, A. Choromanska, L. Sagun among others), as well as
recent work of E. Subag and E.Subag and O.Zeitouni.

In the early '90s, Gromov introduced a notion of hyperbolicity for
geodesic metric spaces. The study of groups of isometries of such
spaces has been an underlying theme in much of the work in geometric
group theory since that time. Many geodesic metric spaces, while not
hyperbolic in the sense of Gromov, nonetheless display some
hyperbolic-like behavior. I will discuss a new invariant, the Morse
boundary of a space, which captures this behavior. (Joint work with
Harold Sultan and Matt Cordes.)

We consider the following problem. Does there exist an absolute constant C such that for every natural number n, every integer 1 \leq k \leq n, every origin-symmetric convex body L in R^n, and every measure \mu with non-negative even continuous density in R^n, \mu(L) \leq C^k \max_{H \in Gr_{n-k}} \mu(L \cap H}/|L|^{k/n}, where Gr_{n-k} is the Grassmannian of (n-k)-dimensional subspaces of R^n, and |L| stands for volume? This question is an extension to arbitrary measures (in place of volume) and to sections of arbitrary codimension k of the hyperplace conjecture of Bourgain, a major open problem in convex geometry. We show that the above inequality holds for arbitrary origin-symmetric convex bodies, all k and all \mu with C \sim \sqrt{n}, and with an absolute constant C for some special class of bodies, including unconditional bodies, unit balls of subspaces of L_p, and others. We also prove that for every \lambda \in (0,1) there exists a constant C = C(\lambda) so that the above inequality holds for every natural number, every origin-symmetric convex body L in R^n, every measure \mu with continuous density and the codimension of sections k \geq \lambda n. The latter result is new even in the case of volume. The proofs are based on a stability result for generalized intersections bodies and on estimates of the outer volume ratio distance from an arbitrary convex body to the classes of generalized intersection bodies.

In this talk, I will survey the recent understandings on
the motion of water waves obtained via rigorous
mathematical tools, this includes the evolution of smooth initial
data and some typical singular
behaviors. In particular, I will present our recently results on
gravity water waves with angled crests.